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4 Answers
4

First of all, moves 2 and 4 absolutely cannot be swapped for the same result. If white plays at 4 first, black would connect at 2. And white has to spend another move protecting the cut. It's a whole world of difference.

Secondly, the name for white 2 is "throw-in". The corresponding Chinese name is "扑", which means "to leap".

In the actual game, black presumably captures two stones (2 and 6) while white also captures two stones, so that part balances out. Compared to the optimal solution (lower figure), black put 2 more stones in his own territory (at a and b in the lower figure) while white only put one more (c). So that's a -1 point for black right there.

The true loss for black lies in the fact that black could have had sente (lower figure), instead of gote as happened in the actual game. The point value of that is dependent on the rest of the board, but is usually quite high.

Now that the black mistake in the original sequence has been explained by Laval, maybe it's interesting to give an analyse of the original position. It's not very easy to tell who will be able to play first at a.

If the temperature of the board is sufficiently low (<6 pts) then a is a double sente and should be played as soon as possible by both players. (Net gain will be 1 point if black is ko master, but up to 3 points if he is not)

However let's have a look at the continuations if the other player does not reply:

Black continuation is a lot bigger than white's one. (Sorry in my previous count, I miscounted.)

Black continuation is only slightly bigger than white's one. This means that both players might have the opportunity to play at a first. Black might play it while the temperature of the board is between 7 and 6 (that is, before it strictly becomes double sente), but white will only play it below 6. So, black having the opportunity to play a first is a little bit more probable, and it might be taken into account when estimating the score.

Interesting point. Looks like you are going off of mathematical Go, which is known to a very good, if counter-intuitive algorithm for endgames. However, please do remember that the move at A2 has intrinsic value because after white plays there, black has lost a point on the left, and has to spend an extra move after capturing the white stone to avoid losing more points. Ignoring any follow-ups, the move is worth about two and a half points single sente on its own. Normally, it's hard to find a single-sente move larger than A2 in the endgame phase. I suppose it is possible, though.
– LavalNov 13 '11 at 17:58

-Black lost 2 points, one at 5 and another one that is needed to be played at 2.
-White lost 2 points due to the reduction of 1 and 4. However white will need to defend and connect 4 - this will cost white 1 more point.

So total win/loss for this is:
Black -2 points.
White -3 points.

-Note black did not lose a point due to 2 and 3. As the exchange resulted in a equalization of points, 1 stone gained (1 point) for 1 point lost (due to the capture).

After 5 white capture two stones by playing again at 2 (followed by black recapturing). Eventually black may fill at 2, but then white can connect at 3, this part of the exchange doesn't earn or lose any point for either player though, so it is essentially a dame exchange.
– MefNov 13 '11 at 19:58

The "slow" connection on A2 is actually the biggest move. That's because it protects the cut at c2, as well as the stone on a3 from capture. And it threatens a big sequence starting with a4, meaning that White is forced to play a4, and Black has the next move.

Compared with the diagrams above, Black is two points better, and White is one point better, for a net gain of one point for Black, who keeps sente.

In a close game, if one person gets to make 5-10 one-point moves in sente, that's enough to win the game.

So the above sequence is better than Black's pushing out at a4, getting cut by a2, and having to capture on a1 AND go back to prevent the cut at c2. Even if it were a point more than the sequence above, it would be gote.